For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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1answer
779 views

Singular Value Decomposition of a block diagonal matrix

For a block diagonal matrix, we have an identity for its cholesky decomposition i.e. $chol(Z) = chol(blockdiag(A,B,...)) = blockdiag(chol(A),chol(B),...)$ (Here, $Z = blockdiag(A,B,...)$) I want to ...
1
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2answers
226 views

A number of SVD components; understanding the relation

Related to my work is the concept of Singular Value Decomposition (SVD). Namely, given some matrix $B\in\mathbb{R}^{n\times m}$, $n\geq m$, SVD can be written as $$B=U\Sigma V^T,$$ where ...
3
votes
2answers
162 views

transitivity of commutator

I remember a quantum mechanics lecture where my professor said "Two matrices $A, B$ which commute with a third matrix $C$, $[A,C]=[B,C]=0$, commute with each other: $[A,B]=0$." I pointed out the ...
2
votes
1answer
105 views

Ways of computing $A^\infty$

As a follow-up on this question, I would like to ask which one is the better way of computing $A^\infty = \lim_{n \rightarrow \infty} A^n$. Repeatedly square, computing $A^2, A^4, A^8$ and so on. Do ...
3
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0answers
114 views

When is a matrix congruent to a diagonal matrix and how to find the congruent transformation?

What matrix can be congruent to a diagonal matrix and how can we find the congruent transform and the diagonal matrix? One special case is when the congruence is also similarity. For example, for a ...
0
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1answer
50 views

Problem related to a complex matrix

I am stuck on the following problem: Let $P$ be a $2 \times 2$ complex matrix such that trace $P=1$ and $\det P=-6.$ Then trace $(P^4-P^3)=?$ Can someone point me in the right direction? ...
3
votes
1answer
658 views

“Inverse” of tensor product

I am trying to figure out something. I have a 4-tensor $\phi_{i \, j \, k \, \ell}$ and I know that $\phi = A \otimes B$, being $A$ and $B$ two matrices. With indices, I know this: $\phi_{i \, j \, k ...
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1answer
538 views

How to combine covariance matrices?

I have a data set of points in three dimensions. I'm calculating the barycenter (mean) and $3\times3$ covariance matrix from this data set. I store the average, the $3\times3$ matrix (where really ...
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2answers
45 views

Solve a written problem with matrix

I have the following problem described here: The government attributes an allocation to the children who benefits child-care services. The children are splitted inside 3 groups: preschool, first ...
0
votes
2answers
57 views

matrix question - help needed

I am doing revision on matrices and came across this question. The solution (the matrix provided below the question) is there. I am not sure how or why 180 is in the position (1,4) (row and column ...
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1answer
320 views

How can one define the trace of a linear operator on any finite dimensional vector space, using the fact that $tr(A) = tr(P^{-1}AP)$?

Firstly, I had to prove that $tr(AB) = tr(BA)$ and deduce that the trace is an invariant of similarity i.e. that $tr(A) = tr(P^{-1}AP)$ for any $A$ and invertible $P$. I could prove the first part - ...
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0answers
60 views

Convert hermitian matrix to symmetric

Is there some simple transformation (or a simple way to find it) which would convert any given hermitian matrix $A$ to a symmetric matrix $B$ with the same spectrum as that of $A$ (so I guess that ...
2
votes
1answer
79 views

Handling matrix of differential operator when using the Ritz method for an extremum problem

The internal energy (or strain energy) of a typical structural member under a displacement field $\{u\}$ can be represented as: $$ U = \frac{1}{2}\int_{V}[u]^T[B]^T[F]^T[B][u]dV $$ where $V$ is the ...
4
votes
0answers
173 views

Invariant of matrix under elementary transformations

$\DeclareMathOperator{\rank}{rank}$ Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix $$ B = \begin{bmatrix} A & b \\ b^T & c ...
5
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2answers
254 views

Can a Gaussian integer matrix have an inverse with Gaussian integer entries?

Is there any way to characterize the set of complex matrices with Gaussian integer entries whose inverses also have Gaussian integer entries? I'm aware of the numerous examples of integer matrices ...
1
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2answers
159 views

Linear Transformations: Scaling along the line $y=x$

What is the geometric meaning of scaling an object by a factor $k$ along the line $y=x$? What will be the shape of a square with vertices $(2,1)$, $(3,2)$, $(3,1)$, and $(2,2)$ if it is scaled by a ...
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1answer
476 views

Trace inequality for real matrices

Is there any general result characterizing real matrices $A$ such that $$[\mathrm{tr}(A)]^2\leq n\mathrm{tr}(A^2)?$$ I can see that the inequality holds if: all eigenvalues of $A$ are real (by the ...
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1answer
156 views

Cancellation property in matrices.

I just found a question which is based on a doubt I have carrying for over 10 years. If $ACC^t=BCC^t$ : $C^t$ means transpose of $C$ Is $A=B$ $AC=BC$ Sorry if this is a trivial question. ...
0
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1answer
97 views

Prove that $(X'X)^{-1}X'AX(X'X)^{-1}-(X'A^{-1}X)^{-1}$ is positive definite

How to prove if A is a positive definite matrix, then $(X'X)^{-1}X'AX(X'X)^{-1}-(X'A^{-1}X)^{-1}$ is also positive definite? Here $X'$ denotes the transpose of $X$. $A$ is square and $X$ is $n\times ...
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0answers
58 views

When is a matrix called well-ordered?

We have a quick question, looking for information and/or references links. Is there a more specific mathematical definition/criteria for when matrices can be called well ordered or totally ordered? ...
0
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1answer
36 views

Name of type of Matrix

I am struggling to remember the name of this type of Matrix. Could anyone assist? $$ \begin{bmatrix} 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0 ...
5
votes
3answers
439 views

Trace of an integral

Given appropriate matrices $A$ and $B_x$, is $\,\,tr\left(\int A B_x dx\right) = \int tr\left(A B_x\right) dx\,\,?$ If so, is it true by the argument that it transfers from (discrete) sums?
2
votes
1answer
81 views

On triangular decomposition of square matrix

Let $L\in Gl_n(\mathbb{C})$ and define $A=LL^*$. Let us consider another decomposition such as $A=L_1L_1^*$. What is the relation between $L$ and $L_1$. One obvious relation is $L_1=LU$ where $U$ is ...
2
votes
1answer
544 views

Limit of matrix powers.

Consider an arbitrary matrix $A$ with eigenvalues within the unit circle. Is there a nice formula for $A^\infty = \lim_{n \rightarrow \infty} A^n$? In particular, maybe there is a formula which ...
2
votes
2answers
323 views

The relationship between eigenvalues of matrices $XY$ and $YX$

If $X \in \mathbb{C}^{m \times n}$ and $Y \in \mathbb{C}^{n \times m}$ ($m \geq n$), how to prove that $\lambda (XY) = \lambda (YX) \cup \underbrace{\left \{ 0, ..., 0 \right \}}_{m-n}$? Here, ...
1
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1answer
67 views

Linear algebra, Schur set

Can you guys give me some hints on how I can start this problem? Thanks in advance! Let $ U(n) \subseteq M_n(\mathbb C) $ be the set (group) of all $ n \times n $ unitary matrices. Let $ T ...
0
votes
2answers
103 views

A matrix is normal, if and only if?

Let $A \in M_n(\mathbb C)$. Let $\langle \; \cdot\; , \; \cdot\; \rangle$ be the standard inner product in $ \mathbb C^n$, viewed either as row vectors or as column vectors. Let $r_j$ be the $j$-th ...
0
votes
1answer
28 views

Expressing a transformation matrix

Let $B=\{v_1,...,v_n\}$ and $C=\{w_1,...,w_n\}$ be bases to $V$. Suppose: $w_i=m_{i1}v_1+...+m_{in}v_n$ for $m_{ij}\in F, 1\le i,j \le n$. $M$ is an invertible matrix whose ($i,j$) member is $m_{ij}$. ...
2
votes
2answers
62 views

Is the linear dependence test also valid for matrices?

I have the set of matrices $ \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} $ $ \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ ...
1
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1answer
241 views

How would I find this eigenvalue?

I'm told to let $A$ be the matrix of the linear transformation $T$ and without writing $A$, find an eigenvalue of $A$ and describe the eigenspace. The first is to let $T$ be the transformation on ...
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1answer
85 views

Condition number of a $9\times9$ matrix

would like someone to look over this and assure me I'm not making a silly mistake.... Given a $3\times9$ matrix $V$: $$ \small\begin{bmatrix} 1.0814 & -0.1251 & -0.1726 & -1.4443 & ...
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votes
2answers
48 views

Quick question about proofs of theorem concerning Jordan basis

I have a question about proofs of this theorem: Let $K$ be an algebraically closed field, $V$ be a finite-dimensional space over $K$ and $f : V → V$ be a linear operator. Then there exists a Jordan ...
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1answer
696 views

Linear Algebra: Least-Squares Approximation & “Normal Equation”

I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] ...
1
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1answer
211 views

How to prove that $D := ABC$ is also positive definite?

Let $A,$ $B$ and $C$ be symmetric, positive definite matrices and suppose that $D := ABC$ is symmetric. How might I prove that $D$ is also positive definite?
2
votes
1answer
67 views

I need to diagonalize this matrix but I'm not sure it can be

This is the matrix I need to diagonalize: $A=\left[\begin{matrix}3&2\\0&3\end{matrix}\right]$. So I found the eigenvalue by taking the determinant of $(A-\lambda I)$ and solving for ...
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2answers
6k views

Matlab function, rotation matrix

Is this the correct way to calculate a rotation matrix for a given angel around a unit vector, i am having problems verifying it. ...
1
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1answer
147 views

how to calculate the “variance OF the covariance” matrix : E[vech(x x') vech(x x')'] for normal distributed x?

Supposing a vector x follows normal distribution. I want to calculate the expectation of the "variance Of the covariance matrix" (not variance-covariance matrix) in a vector form, meaning E[vech(x ...
2
votes
2answers
132 views

Picard iterations of a matrix

I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one. We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
3
votes
0answers
270 views

Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
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votes
2answers
53 views

Matrix Algebra (Elementary)

I have $\hat\xi =\lambda_1\textbf{1V}^{-1} + \lambda_2\textbf{rV}^{-1}$ and sub it in to my two constraints, namely, $\xi\textbf{1}^T = 1$ and $\xi\textbf{r}^T = \mu$. My lecture notes then say set ...
21
votes
4answers
844 views

Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times ...
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1answer
2k views

Get code words from generator matrix

I have some issue regarding the generator matrix. Please can some body can explain me "How to get Codebook from Generator matrix?" Following is my issue Generator matrix has 3 code words. Then ...
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2answers
68 views

Identifying matrix vector multiplication

I have the following question in a book: According to the book, the answer is (D). But I don't understand how. Isn't this just scalar multiplication? The solution in the book says that I have to ...
3
votes
1answer
89 views

Find the smallest square matrix in which some objects fit following some rules

I have to put some objects in a matrix. The data of these objects is given in another matrix in which each line contains an object, and the first column represents its width, and the second its ...
0
votes
2answers
221 views

Volume of a parallelepiped, given three vectors

I want the volume of a parallelpiped and I have the three vectors $$4e_1+2e_2-e_3$$$$e_1-3e_2-2e_3$$$$2e_1-e_2+3e_3$$ that coinciding with three of the parallelpipeds sides. HON-base I made it into a ...
2
votes
6answers
550 views

Let $A^{27}=A^{64}=I$, show that $A=I$

Let $A$ be a square matrix, $A^{27}=A^{64}=I$, show that $A=I$
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1answer
433 views

Is the sum of two singular matrix also a singular matrix

If $A$ and $B$ are singular and both $n\times n$, is $A+B$ also singular?
3
votes
4answers
181 views

Is there a good intuitive way to understand why matrix B is inverse of A when matrix A|I is turned into I|B

I'm looking for some help with my intuition of basic matrix operations, specifically finding a matrix's inverse (as per my subject line). I have no problems with the steps. The basic row operations ...
1
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1answer
56 views

finding the P matrix (diagonalization of a matrix)

I'm trying to find the diagonalization of a matrix : this is my matrix : $$ A =\begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 1 & 1 & 1 \\ ...
7
votes
0answers
456 views

Condition of an eigenvector problem

Please, somebody help me with this problem. [Ciarlet 2.3-5] Let ${A}$ and ${B} = {A} + \delta{A}$ be two symmetric matrices with eigenvalues $$\alpha_1\ \leq\ \alpha_2\ \leq\ \ldots\ \leq\ ...